Theorem xpdisj1 5178

Description: Cross products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)

Assertion

Ref	Expression
xpdisj1	|- ( ( A i^i B ) = (/) -> ( ( A X. C ) i^i ( B X. D ) ) = (/) )

Proof of Theorem xpdisj1

Step	Hyp	Ref	Expression
1		inxp 4880	. 2 |- ( ( A X. C ) i^i ( B X. D ) ) = ( ( A i^i B ) X. ( C i^i D ) )
2		xpeq1 4754	. . 3 |- ( ( A i^i B ) = (/) -> ( ( A i^i B ) X. ( C i^i D ) ) = ( (/) X. ( C i^i D ) ) )
3		0xp 4821	. . 3 |- ( (/) X. ( C i^i D ) ) = (/)
4	2, 3	eqtrdi 2281	. 2 |- ( ( A i^i B ) = (/) -> ( ( A i^i B ) X. ( C i^i D ) ) = (/) )
5	1, 4	eqtrid 2277	1 |- ( ( A i^i B ) = (/) -> ( ( A X. C ) i^i ( B X. D ) ) = (/) )

Syntax hints:   -> wi 4   = wceq 1398   i^i cin 3209   (/)c0 3505   X. cxp 4738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4221  ax-pow 4279  ax-pr 4314

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3506  df-pw 3667  df-sn 3688  df-pr 3689  df-op 3691  df-opab 4165  df-xp 4746  df-rel 4747

This theorem is referenced by:  djudisj  5181  xp01disjl  6658  xpfi  7183  djuinr  7345